An element-free IMLS-Ritz method for numerical solution of three-dimensional wave equations

This paper presents an element-free based numerical approach for solving three-dimensional wave equations based on the Ritz minimization procedure. The method involves the use of a set of orthogonal shape functions to approximate its field variables. In this study, an improved moving least-squares (IMLS) is used to generate sets of orthogonal shape functions that reduced the number of unknown coefficients in the trial functions. The entire approximation procedure can be easily implemented numerically. The accuracy of the approximation can be enhanced by increasing the number of nodes used in the computation. As the result of the above procedures, a final algebraic equation system is derived through discretizing the constructed functional. The functional is established by enforcing the Dirichlet boundary conditions via the penalty approach. The simplicity and applicability of the element-free method are demonstrated by solving several selected linear and nonlinear three-dimensional wave equations. Besides the convergence study, the present results are compared with the available published solutions from the literature, where possible, verifying the accuracy, efficiency and reliability of the IMLS-Ritz method.